Geometric Perron-Frobenius theory in finite dimensions Hans - - PowerPoint PPT Presentation

Geometric Perron-Frobenius theory in finite dimensions

Hans Schneider Chemnitz October 2010

geompfchmn 2010.09.08, 11:05 September 8, 2010 Hans Schneider Geometric Perrron-Frobenius 1 / 14

cones Definition K a cone in Rn

1

K +K ⊆ K (x,y ∈ K = ⇒ x +y ∈ K)

2

R+K ⊆ K (α ≥ 0,x ∈ K = ⇒ αx ∈ K

3

K pointed K ∩K = {0} (x,−x ∈K= ⇒ x = 0)

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cones Definition K a cone in Rn

1

K +K ⊆ K (x,y ∈ K = ⇒ x +y ∈ K)

2

R+K ⊆ K (α ≥ 0,x ∈ K = ⇒ αx ∈ K

3

K pointed K ∩K = {0} (x,−x ∈K= ⇒ x = 0) We assume K closed (in Euclidean topology of Rn)

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proper cones Definition cone K proper: span(K) = K −K = Rn (∀z ∈ Rn ∃x,y ∈ K, z = x −y)

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proper cones Definition cone K proper: span(K) = K −K = Rn (∀z ∈ Rn ∃x,y ∈ K, z = x −y) proper cone p.o’s Rn y ≥ x : y −x ∈ K

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proper cones Definition cone K proper: span(K) = K −K = Rn (∀z ∈ Rn ∃x,y ∈ K, z = x −y) proper cone p.o’s Rn y ≥ x : y −x ∈ K x ≥ 0 ⇐ ⇒ x ∈ K x > 0 ⇐ ⇒ x ∈ int K

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faces Definition F a face of cone K −−F K: F is a subcone of K and x ∈ F,0 ≤ y ≤ x = ⇒ y ∈ F

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faces Definition F a face of cone K −−F K: F is a subcone of K and x ∈ F,0 ≤ y ≤ x = ⇒ y ∈ F F a proper face of K: F = 0, F = K

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three examples Rn

+

Faces: (+,0,+0,...,0,+)

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three examples Rn

+

Faces: (+,0,+0,...,0,+) (x2

1 +x2 2 +···+x2 n−1)(1/2) ≤ xn

Proper faces: (x2

1 +x2 2 +···x2 n−1) ≤ xn

Extremals

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three examples Rn

+

Faces: (+,0,+0,...,0,+) (x2

1 +x2 2 +···+x2 n−1)(1/2) ≤ xn

Proper faces: (x2

1 +x2 2 +···x2 n−1) ≤ xn

Extremals (Real) Space: Hermitian n ×n matrices Cone: Positive semi-definite matrices Faces: Matrices simult similar to 0k ⊕X

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nonnegative operators K a proper cone in Rn A ≥ 0 : AK ⊆ K A > 0 : AK ⊆ int (K)

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nonnegative operators K a proper cone in Rn A ≥ 0 : AK ⊆ K A > 0 : AK ⊆ int (K) F an invariant face: AF ⊆ F

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nonnegative operators K a proper cone in Rn A ≥ 0 : AK ⊆ K A > 0 : AK ⊆ int (K) F an invariant face: AF ⊆ F A ≥ 0 irreducible: A leaves no proper face of K invariant: ⇐ ⇒ (I +A)n−1 > 0

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Aim of talk The aim is to relate the structure of invariant faces of the cone to the spectral properties of an operator nonnegative w.r.t. the cone.

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Aim of talk The aim is to relate the structure of invariant faces of the cone to the spectral properties of an operator nonnegative w.r.t. the cone. As we related the graph theoretic and spectral properties

• f a nonnegative matrix both in classical nonneg alg and

max alg.

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fin dim Krein-Rutman 1940’s Theorem A ≥ 0 irreducible spec rad ρ(A) ∈ spec(A) ρ(A) simple eigenvalue ∃!x > 0, Ax = ρx y 0,Ay = λy = ⇒ λ = ρ, y = x

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fin dim Krein-Rutman 1940’s Theorem A ≥ 0 irreducible spec rad ρ(A) ∈ spec(A) ρ(A) simple eigenvalue ∃!x > 0, Ax = ρx y 0,Ay = λy = ⇒ λ = ρ, y = x And for reducible matrices?

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RECALL combinatorial theory ρi := ρ(Aii) i is a distinguished vertex: i

∗

← j = ⇒ ρi > ρj Theorem Let A be a nonnegative matrix in FNF. Then the nonnegative eigenvectors of A correspond to the distinguish vertices of A: for for each distinguished vertex i of ∆(A) there is nonnegative eigenvector xi with Axi = ρixi such that xi

j > 0

if i

∗

← j xi

j = 0

• therwise

These are linearly independent, and all others are nonneg lin combs.

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distinguished faces F K : F an invariant face of K ρF = ρ(A|F) F K distinguished: F,G invariant, G ⊳F = ⇒ ρG < ρF Theorem K a proper cone in Rn. Each distinguished face contains an eigenvector of ρF in its relative interior. For any particular eigenvalue, the eigenvectors so

• btained are the extremals of the cone of nonnegative

eigenvectors.

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chains of semi-dist inv faces F K is semi-distinguished : G F, G invariant = ⇒ ρG ≤ ρF

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chains of semi-dist inv faces F K is semi-distinguished : G F, G invariant = ⇒ ρG ≤ ρF A ∈ Rn×n

+

in FNF indλ(A) : = max size of J–block for λ = min{k : N = N (λI −A)k+1 = N (λI −A)k}

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chains of semi-dist inv faces F K is semi-distinguished : G F, G invariant = ⇒ ρG ≤ ρF A ∈ Rn×n

+

in FNF indλ(A) : = max size of J–block for λ = min{k : N = N (λI −A)k+1 = N (λI −A)k} Invariant Fλ-face: ρF = λ K F1 ⊲F2 ⊲···⊲F : proper chain of inv semi-dist λ faces

• rdλ = max length of such a chain of faces

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generalization of Rothblum’s theorem Tam-S (2001) Theorem indλ ≤ ordλ If K is polyhedral (fin gen), indρ = ordρ

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generalization of Rothblum’s theorem Tam-S (2001) Theorem indλ ≤ ordλ If K is polyhedral (fin gen), indρ = ordρ If K = Rn

+ this is a generalization by Hershkowitz-S of

Rothblum’s theorem (case λ = ρ)

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Tam-S, Matrices leaving a cone invariant , Art. 26, Handbook of Linear Algebra (ed. L. Hogben) 2007

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Tam-S, Matrices leaving a cone invariant , Art. 26, Handbook of Linear Algebra (ed. L. Hogben) 2007 THANK YOU And please allow me one more slide!

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